Large Relativistic Corrections to Nonrelativistic $M1$ Transitions in Heavy Quarkonium

TL;DR

This paper uses the relativistic Bethe-Salpeter equation to calculate electromagnetic transitions in heavy quarkonium, revealing large relativistic corrections even in bottomonium, which significantly impact decay rate predictions.

Contribution

It introduces a comprehensive relativistic calculation of M1 and higher multipole transitions in heavy quarkonium, extending beyond traditional nonrelativistic approaches.

Findings

Relativistic corrections in charmonium decays range from 68.1% to 83.2%.

Relativistic corrections in bottomonium decays range from 65.9% to 75.2%.

Higher multipole effects are significant in heavy quarkonium transitions.

Abstract

As double heavy quarkonia, charmonium and bottomonium are generally considered to have small relativistic corrections and can be treated using nonrelativistic models. However, this is not always the case. In this paper, we employ the relativistic Bethe-Salpeter (BS) equation method to calculate the electromagnetic (EM) radiative decays of heavy quarkonium where the $M1$ transition provides the leading-order contribution. Compared to nonrelativistic method which only computes $M1$ transition, our calculations include $M1+E2+M3+E4$ transitions, where the higher-order multipoles, $E2$, $M3$, and $E4$, account for relativistic corrections. The study finds that relativistic effects are large in such transitions even for bottomonium. For instance: the relativistic corrections in the decays $\psi(nS)\rightarrow\gamma\eta_c(mS)$ ($n\geq m$) range from $68.1\%$ to $83.2\%$, while those for $\Upsilon(nS)\rightarrow\gamma\eta_b(mS)$ range between $65.9\%$ and $75.2\%$.